Not much that I've seen around Ouray, but there are trails near Ridgway. A fellow who has a bike shop there posts here sometimes. The 4x4 roads would be a nightmare on a bike - think sitting in the granny gear for a couple of hours with jeeps passing inches away every 10 seconds.
There are some good trails up on the Uncompahgre Plateau that would take maybe an hour to get to. Telluride is about an hour away and Junction, Gunnison, Crested Butte and Durango are within a couple of hours.